
In this case, the MAP estimation \(f^*(X) = 2*1_{|X - \mu _1|\ge |X-\mu _{-1}|}-1\)  and the minimal Bayesian mis-classification error is \(\Phi (-\sqrt{\rho }/2)\)  where \(\Phi \)  denotes the cumulative standard Gaussian distribution function. For more general cases where
the two classes are associated with sub-Guassian distributions \(\mathbb {P}_{Y}\) , s.t. \(\mathbb {P}_{Y}(|X| > t) \in [c_1\exp (- c_2t^2),C_1\exp (- C_2t^2)]\) , for some non-negative constants \(c_1,c_2,C_1,C_2\) , a similar connection between \(\xi ^*\)  and \(\rho \)  can be shown by leveraging sharp sub-Gaussian lower bounds [1]}. We will specify the connection to SNR in our case in Sec.  and the SNR \(\rho \)  will be used as the main bridge to compare the mis-classification errors of non-linear v.s. linear models.
